Question

let A (1 , 2, -3), B (2, 1, 4) and C (0, 0, 2) be three points in R^3

a) give the parametric equation of the line orthogonal to the plane containing A, B and C and passing through point A.

b) find the area of the triangle ABC

linear-algebra question

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Answer #1

Let A = (−5,3,5), B = (−10,0,2), C = (−5,0,−3), and D =
(0,3,0).
a) Find the area of the parallelogram determined by these four
points.
b) The area of the triangle ABC
c) The area of the triangle ABD.

1. A plane passes through A(1, 2, 3), B(1, -1, 0) and
C(2, -3, -4). Determine vector and parametric equations of
the plane. You must show and explain all steps for full marks. Use
AB and AC as your direction vectors and point A as your starting
(x,y,z) value.
2. Determine if the point (4,-2,0) lies in the plane with vector
equation (x, y, z) = (2, 0, -1) + s(4, -2, 1) + t(-3, -1,
2).

1) Let a = 〈4,−5,−2〉 and b = 〈2,−4,−5〉 Find the projection of b
onto a. proj a b=
2) Find the area of a triangle PQR, where P=(−2,−4,0),
Q=(1,2,−1), and R=(−3,−6,5)
3) Complete the parametric equations of the line through the
points (7,6,-1) and (-4,4,8)
x(t)=7−11
y(t)=
z(t)=

Let P be the plane given by the equation 2x + y − 3z = 2. The
point Q(1, 2, 3) is not on the plane P, the point R is on the plane
P, and the line L1 through Q and R is orthogonal to the plane P.
Let W be another point (1, 1, 3). Find parametric equations of the
line L2 that passes through points W and R.

Let a = 2i -3k; b= i+j-k
1) Find a x b.
2) Find the vector projection of a and b.
3) Find the equation of the plane passing through a with normal
b.
4) Find the equation of the line passing through the points a and
b.
Please help and show your steps. Thank you in
advance.

4)
Consider the polar curve r=e2theta
a) Find the parametric equations x = f(θ), y =
g(θ) for this curve.
b) Find the slope of the line tangent to this curve when
θ=π.
6)
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the...

3. Consider the plane with a normal vector 〈2, 5, −1〉 which
contains the point (3, 5, −1), and the plane containing the points
(0, 2, 1), (−1, −1, 1), and (1, 2, −2). Determine whether the
planes are parallel, orthogonal, or neither.

Let A = [1 5 ; 3 1 ; 2 -4] and b = [1 ; 0 ; 3] (where semicolons
represent a new row)
Is equation Ax=b consistent?
Let b(hat) be the orthogonal projection of b onto Col(A). Find
b(hat).
Let x(hat) the least square solution of Ax=b. Use the formula
x(hat) = (A^(T)A)^(−1) A^(T)b to compute x(hat). (A^(T) is A
transpose)
Verify that x(hat) is the solution of Ax=b(hat).

6) please show steps and explanation.
a)Suppose r(t) = < cos(3t), sin(3t),4t
>.
Find the equation of the tangent line to r(t)
at the point (-1, 0, 4pi).
b) Find a vector orthogonal to the plane through the points P
(1, 1, 1), Q(2, 0, 3), and R(1, 1, 2) and the area of the triangle
PQR.

Find the area of a triangle with vertices A (1, 4, -1), B (-1, 2,
0), C (1, 1, 3).

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